JEF Paper Coding (2022) - v3.3 Double Sort & Quintiles (For Monthly Data - Size & Investment)

Size-Operating profitability portfolios are formed based on the NYSE, NASDAQ and AMEX.

Further improvement: + Change the risk-free rate data to the ones compatible with different frequency requirements.

Data cleaning and Preparation

Information about initial coding setup:

1. freq sets the data frequency for the following analysis, 12 for monthly data, 4 for quarterly data and 1 for annual data.
2. start.ym gives the earliest reasonable starting point of the series, which is January 1966, based on the available number of firms in the data set.
3. After the preliminary data cleaning, port_market is the market portfolio data (including NYSE, NASDAQ and AMEX), ports_all contains different deciles. All the data are stored in the file named as market.names and data.names. I’ve finished creating the measures based on characteristic deciles, so I’ll have a close look at your results shortly. The decile data is attached. As mentioned, these are based on a single characteristic sort, which will hopefully provide new insight into characteristic based predictability. The characteristics are as follows:

• Size (size)

• Book-to-market ratio (bm)

• Operating profitability (op)

• Investment (inv); and

• Momentum (mom)

  The filenames indicate which is which, and the quantiles are denoted from Q1 (low value on the characteristic) to Q5 (high value on the characteristic) with the breakpoints identified according to the NYSE distribution. The basis of the sorts are consistent with Fama and French (2015) (and their earlier work) in that re-sorting occurs only one a year. An alternative is to re-sort monthly, which I’ve done for other papers applying a similar approach, but doing so comes with its own drawbacks.

4. RF denotes the risk-free rate, which is the average of the bid and ask.

Notes: Seems that the big-value and small-growth portfolios include less firms comparing the other four characteristic portfolios, around half of them.

Figure 1 - Log Cumulative Index

Log cumulative realised portfolio return components for seven portfolios - the market portfolio and six size and book-to-market equity ratio sorted portfolios. All following figures demonstrate the monthly realised price-earnings ratio growth (gm), earnings growth (ge), dividend-price (dp) and the portfolio return index (r) with the values in January 1966 as zero for all portfolios.

.

Table 1 - Summary statistics of returns components

The correlations between gm and ge might be a bit too high comparing to Ferreira and Santa-Clara (2011). Need to check the code again.

Need to go back to the construction process of Prof Robert Shiller’s CAPE.

‘kable’ for Table Creation

## Warning: `columns = vars(...)` has been deprecated in gt 0.3.0:
## * please use `columns = c(...)` instead

## Warning: `columns = vars(...)` has been deprecated in gt 0.3.0:
## * please use `columns = c(...)` instead

Table 1 - Summary statistics of returns components
monthly data starts from Jan 1967 and ends in Dec 2019.
	Panel A: univariate statistics								Panel B: Correlations
	Mean	Median	SD	Min	Max	Skew	Kurt	AR(1)	gm	ge	dp	r
Market
gm	0.02	-0.03	3.12	-15.26	13.28	-0.19	4.42	0.92	1.00	-0.51	-0.03	0.07
ge	0.76	1.11	5.34	-22.01	19.34	-0.50	2.44	0.33	-0.51	1.00	-0.03	0.81
dp	0.28	0.27	0.09	0.09	0.50	0.14	-0.70	0.98	-0.03	-0.03	1.00	-0.03
r	0.94	1.25	4.43	-22.48	16.58	-0.51	1.85	0.05	0.07	0.81	-0.03	1.00
B_Q1
gm	0.00	-0.06	7.20	-55.83	32.18	-0.88	8.78	0.68	1.00	-0.81	-0.08	0.12
ge	0.78	0.74	8.11	-28.00	58.66	0.60	5.07	0.46	-0.81	1.00	0.03	0.48
dp	0.32	0.30	0.15	0.11	1.06	1.03	1.17	0.93	-0.08	0.03	1.00	-0.05
r	1.09	1.31	4.71	-22.00	20.29	-0.30	1.88	0.05	0.12	0.48	-0.05	1.00
B_Q2
gm	0.05	-0.06	2.72	-16.17	9.30	-0.62	4.36	0.80	1.00	-0.49	0.04	0.11
ge	0.63	0.76	4.59	-17.06	20.34	-0.10	1.62	0.21	-0.49	1.00	-0.04	0.80
dp	0.32	0.28	0.12	0.11	0.63	0.52	-0.86	0.97	0.04	-0.04	1.00	-0.01
r	1.00	1.13	4.03	-19.04	17.22	-0.32	1.61	-0.02	0.11	0.80	-0.01	1.00
B_Q3
gm	0.05	-0.02	2.33	-9.32	13.23	0.47	3.54	0.79	1.00	-0.37	-0.16	0.14
ge	0.64	0.68	4.54	-21.17	17.32	-0.44	1.97	0.18	-0.37	1.00	0.03	0.86
dp	0.30	0.28	0.11	0.10	0.63	0.55	-0.31	0.98	-0.16	0.03	1.00	-0.03
r	0.96	1.17	4.18	-18.91	15.28	-0.47	1.68	0.02	0.14	0.86	-0.03	1.00
B_Q4
gm	0.03	-0.09	2.22	-13.67	9.88	-0.31	5.35	0.78	1.00	-0.32	-0.16	0.14
ge	0.67	0.83	4.73	-21.91	26.21	-0.11	2.45	0.15	-0.32	1.00	-0.01	0.89
dp	0.24	0.21	0.10	0.07	0.54	0.72	-0.18	0.97	-0.16	-0.01	1.00	-0.06
r	0.94	1.06	4.48	-22.96	19.13	-0.42	2.06	0.05	0.14	0.89	-0.06	1.00
B_Q5
gm	0.06	0.20	2.67	-13.34	8.89	-0.42	1.81	0.83	1.00	-0.33	-0.21	0.12
ge	0.73	1.12	5.85	-30.02	24.44	-0.52	2.00	0.20	-0.33	1.00	0.03	0.89
dp	0.17	0.15	0.08	0.03	0.51	0.78	0.20	0.96	-0.21	0.03	1.00	-0.05
r	0.86	1.23	5.47	-24.54	21.37	-0.36	1.39	0.06	0.12	0.89	-0.05	1.00
S_Q1
gm	-0.35	-0.22	5.64	-71.24	36.80	-3.30	46.37	0.66	1.00	-0.63	0.12	0.03
ge	1.65	1.90	8.46	-41.91	68.99	0.61	9.44	0.39	-0.63	1.00	-0.14	0.75
dp	1.22	0.44	1.80	0.13	9.41	2.45	5.41	0.96	0.12	-0.14	1.00	-0.08
r	1.27	1.63	6.43	-30.94	31.73	-0.21	2.53	0.14	0.03	0.75	-0.08	1.00
S_Q2
gm	0.14	-0.39	12.69	-71.70	160.78	3.03	44.06	0.56	1.00	-0.92	0.05	0.10
ge	1.02	1.13	13.22	-162.59	61.21	-2.95	39.22	0.48	-0.92	1.00	-0.04	0.30
dp	0.54	0.36	0.89	0.12	7.36	6.29	40.78	0.91	0.05	-0.04	1.00	0.00
r	1.32	1.59	5.24	-27.71	23.85	-0.51	2.59	0.12	0.10	0.30	0.00	1.00
S_Q3
gm	0.13	0.14	10.93	-142.05	146.01	0.23	96.91	0.49	1.00	-0.90	0.04	0.08
ge	1.05	1.08	11.72	-148.38	146.46	-0.43	78.51	0.37	-0.90	1.00	-0.04	0.37
dp	0.39	0.30	0.60	0.11	6.57	8.87	83.86	0.97	0.04	-0.04	1.00	-0.01
r	1.32	1.63	5.16	-25.80	26.03	-0.46	2.64	0.11	0.08	0.37	-0.01	1.00
S_Q4
gm	0.01	-0.04	7.81	-94.57	69.37	-2.05	59.51	0.40	1.00	-0.79	-0.09	0.09
ge	1.18	1.42	9.23	-70.71	91.07	0.90	27.86	0.26	-0.79	1.00	0.02	0.54
dp	0.27	0.24	0.15	0.07	0.90	1.51	3.12	0.94	-0.09	0.02	1.00	-0.07
r	1.22	1.50	5.64	-28.28	27.09	-0.45	2.59	0.13	0.09	0.54	-0.07	1.00
S_Q5
gm	0.01	0.66	13.10	-128.24	53.64	-4.00	34.53	0.51	1.00	-0.88	0.01	0.06
ge	1.03	0.23	14.46	-54.98	129.20	2.89	22.93	0.41	-0.88	1.00	-0.04	0.42
dp	0.20	0.17	0.22	0.06	2.94	8.89	91.90	0.84	0.01	-0.04	1.00	-0.05
r	0.86	1.25	6.75	-32.53	28.33	-0.41	1.77	0.13	0.06	0.42	-0.05	1.00
Note: Panel A in this table presents mean, median, standard deviation (SD), minimum, maximum, skewness (Skew), kurtosis (kurt) and first-order autocorrelation coefficient of the realised components of stock market returns and six size and book-to-market equity ratio sorted portfolios. These univariate statistics for each portfolios are presented separately. gm is the continuously compounded growth rate in the price-earnings ratio. ge is the continuously compounded growth rate in earnings. dp is the log of one plus the dividend-price ratio. *r* is the portfolio returns. Panel B in this table reports correlation matrices for all seven portfolios. The sample period starts from Feburary 1966 and ends in December 2019.

Figure 3 - Cumulative OOS R-sqaure Difference and Cumulative SSE Difference

The cumulative OOS R-square figures show the out-of-sample cumulative R-square up to each month from predictive regressions with listed predictors and from the sum-of-the-parts (SOP) method for each portfolio. The cumulative SSE difference plots indicates the out-of-sample performance of each model. These are evaluated by the cumulative squared prediction errors of the NULL minus the cumulative squared predictirion error of the ALTERNATIVE. The NULL model is the historical mean model, while the ALTERNATIVE model is either the predictive regression model or the SOP model. An incresae in the line suggests better performance of the ALTERNATIVE model and a decrease suggests that the NULL model is better.

Several points to note in the coding:

1. The dividend-price ratio (‘DP’ hereafter) is calculated as the log of 1 plus the frequency-adjusted dividend to price ratio, rather than using the annual dividend. As by this return decomposition, the expected amount of dividend payout in each period should be adjusted by the frequency of the data in the analysis. \[ dp_t = \log (1 + \frac{\tilde{D}_t}{P_t}) = \log (1 + \frac{D_t / n}{P_t}) \text{,} \] where \(D_t\) is the annual dividend payment and \(n\) is the data frequency (e.g. \(n = 1\) for annual data and \(n = 12\) for monthly data) and \(\tilde{D}_t\) is the freqency-adjusted dividend payment for period \(t\).

2. The SOP method by Ferreira and Santa-Clara (2011) decomposes the portfolio return into three components, namely the earnings growth, the prie multiple expansion and the next period dividend-price ratio. Here to generate the SOP prediction, we use the rolling mean of past earnings growth as the expected growth of the next period (denoted as ge1). However, there are other choices, such as recursive means in ge2 and ge3.

3. critica.value = TRUE is the option whether to use boostrap method to calculate the MSE-F critical values. This is used in function Boot_MSE.F.

4. The authors should evaluate the significance of the MSE−F statistic by using the theoret- ical distribution derived in McCracken (2007). The bootstrap-based inference (presented in Pages 9-10) can represent a robustness check and moved to an appendix. Further- more, the authors can also include in the main results the related out-of-sample statistic proposed by Clark and West (2007), which follows a standard Normal distribution. Therefore, readjust the Boot_MSE.F function.

5. Column McCracken in Table 2 (line 604) gives the significance of the out-of-sample \(MSE–F\) statistic of McCracken (2007). \(***\), \(**\), and \(*\) denote significance at the 1%, 5%, and 10% level, respectively. Please refer to the Table 4 on P749 in McCracken (2007) with \(k_2 = 1\) and \(\pi = P/R = \frac{\text{Number of out-of-sample forecasts}}{\text{Number of observations used to form the first forecast}} = 1.6\).

• Examples are in P742-743. See detailed notations

## [1] "market_Allfirms.csv"
## [1] "Market"
## ##------ Mon Aug  8 10:36:35 2022 ------##

## Note: Using an external vector in selections is ambiguous.
## ℹ Use `all_of(actual)` instead of `actual` to silence this message.
## ℹ See <https://tidyselect.r-lib.org/reference/faq-external-vector.html>.
## This message is displayed once per session.
## Note: Using an external vector in selections is ambiguous.
## ℹ Use `all_of(cond)` instead of `cond` to silence this message.
## ℹ See <https://tidyselect.r-lib.org/reference/faq-external-vector.html>.
## This message is displayed once per session.
## Note: Using an external vector in selections is ambiguous.
## ℹ Use `all_of(uncond)` instead of `uncond` to silence this message.
## ℹ See <https://tidyselect.r-lib.org/reference/faq-external-vector.html>.
## This message is displayed once per session.
## Note: Using an external vector in selections is ambiguous.
## ℹ Use `all_of(x)` instead of `x` to silence this message.
## ℹ See <https://tidyselect.r-lib.org/reference/faq-external-vector.html>.
## This message is displayed once per session.

## [1] "OOS R Squared: 0.0047"
## [1] "MSE-F: 1.8524"

## Note: Using an external vector in selections is ambiguous.
## ℹ Use `all_of(predictor)` instead of `predictor` to silence this message.
## ℹ See <https://tidyselect.r-lib.org/reference/faq-external-vector.html>.
## This message is displayed once per session.

## [1] "IS R Squared: 0.0106"
## [1] "OOS R Squared: -0.0104"
## [1] "MSE-F: -4.054"
## [1] "IS R Squared: 0.0045"
## [1] "OOS R Squared: -0.0023"
## [1] "MSE-F: -0.9222"
## [1] "IS R Squared: 0.0051"
## [1] "OOS R Squared: -0.0017"
## [1] "MSE-F: -0.658"
## [1] "IS R Squared: 0.012"
## [1] "OOS R Squared: -0.0086"
## [1] "MSE-F: -3.3515"
## [1] "IS R Squared: 1e-04"
## [1] "OOS R Squared: -0.0079"
## [1] "MSE-F: -3.1035"

## [1] "port_B_Q1.csv"
## [1] "B_Q1"
## ##------ Mon Aug  8 10:36:41 2022 ------##
## [1] "OOS R Squared: 0.0034"
## [1] "MSE-F: 1.3618"

## [1] "IS R Squared: 0.0034"
## [1] "OOS R Squared: -0.0265"
## [1] "MSE-F: -10.1805"
## [1] "IS R Squared: 0.0088"
## [1] "OOS R Squared: 0.0062"
## [1] "MSE-F: 2.4825"
## [1] "IS R Squared: 0.0095"
## [1] "OOS R Squared: 0.0076"
## [1] "MSE-F: 3.04"
## [1] "IS R Squared: 0.0041"
## [1] "OOS R Squared: -0.0221"
## [1] "MSE-F: -8.5421"
## [1] "IS R Squared: 0.0025"
## [1] "OOS R Squared: -0.0045"
## [1] "MSE-F: -1.7604"

## [1] "port_B_Q2.csv"
## [1] "B_Q2"
## ##------ Mon Aug  8 10:36:46 2022 ------##
## [1] "OOS R Squared: 0.0072"
## [1] "MSE-F: 2.8856"

## [1] "IS R Squared: 0.009"
## [1] "OOS R Squared: -0.0188"
## [1] "MSE-F: -7.2949"
## [1] "IS R Squared: 0.0032"
## [1] "OOS R Squared: -6e-04"
## [1] "MSE-F: -0.23"
## [1] "IS R Squared: 0.003"
## [1] "OOS R Squared: -5e-04"
## [1] "MSE-F: -0.1928"
## [1] "IS R Squared: 0.0085"
## [1] "OOS R Squared: -0.0154"
## [1] "MSE-F: -5.9741"
## [1] "IS R Squared: 0.0018"
## [1] "OOS R Squared: -0.0141"
## [1] "MSE-F: -5.4927"

## [1] "port_B_Q3.csv"
## [1] "B_Q3"
## ##------ Mon Aug  8 10:36:51 2022 ------##
## [1] "OOS R Squared: 0.0042"
## [1] "MSE-F: 1.6711"

## [1] "IS R Squared: 0.0082"
## [1] "OOS R Squared: -0.0034"
## [1] "MSE-F: -1.3498"
## [1] "IS R Squared: 0.0045"
## [1] "OOS R Squared: 0.0022"
## [1] "MSE-F: 0.8904"
## [1] "IS R Squared: 0.0048"
## [1] "OOS R Squared: 0.0031"
## [1] "MSE-F: 1.234"
## [1] "IS R Squared: 0.0087"
## [1] "OOS R Squared: -0.0011"
## [1] "MSE-F: -0.4461"
## [1] "IS R Squared: 0.002"
## [1] "OOS R Squared: -0.0044"
## [1] "MSE-F: -1.7491"

## [1] "port_B_Q4.csv"
## [1] "B_Q4"
## ##------ Mon Aug  8 10:36:56 2022 ------##
## [1] "OOS R Squared: -0.0025"
## [1] "MSE-F: -1.0047"

## [1] "IS R Squared: 0.0018"
## [1] "OOS R Squared: -0.012"
## [1] "MSE-F: -4.6718"
## [1] "IS R Squared: 0.0049"
## [1] "OOS R Squared: -0.0077"
## [1] "MSE-F: -3.0192"
## [1] "IS R Squared: 0.0056"
## [1] "OOS R Squared: -0.0064"
## [1] "MSE-F: -2.5077"
## [1] "IS R Squared: 0.0022"
## [1] "OOS R Squared: -0.0117"
## [1] "MSE-F: -4.5588"
## [1] "IS R Squared: 5e-04"
## [1] "OOS R Squared: -0.0043"
## [1] "MSE-F: -1.7084"

## [1] "port_B_Q5.csv"
## [1] "B_Q5"
## ##------ Mon Aug  8 10:37:02 2022 ------##
## [1] "OOS R Squared: 0.0068"
## [1] "MSE-F: 2.6926"

## [1] "IS R Squared: 0.0034"
## [1] "OOS R Squared: -0.0119"
## [1] "MSE-F: -4.6375"
## [1] "IS R Squared: 0.007"
## [1] "OOS R Squared: -0.0085"
## [1] "MSE-F: -3.3386"
## [1] "IS R Squared: 0.0078"
## [1] "OOS R Squared: -0.0075"
## [1] "MSE-F: -2.9591"
## [1] "IS R Squared: 0.004"
## [1] "OOS R Squared: -0.0124"
## [1] "MSE-F: -4.8255"
## [1] "IS R Squared: 8e-04"
## [1] "OOS R Squared: -0.0046"
## [1] "MSE-F: -1.8162"

## [1] "port_S_Q1.csv"
## [1] "S_Q1"
## ##------ Mon Aug  8 10:37:06 2022 ------##
## [1] "OOS R Squared: -0.2479"
## [1] "MSE-F: -78.6571"

## [1] "IS R Squared: 1e-04"
## [1] "OOS R Squared: -0.0402"
## [1] "MSE-F: -15.263"
## [1] "IS R Squared: 2e-04"
## [1] "OOS R Squared: -0.0054"
## [1] "MSE-F: -2.1257"
## [1] "IS R Squared: 7e-04"
## [1] "OOS R Squared: -0.0056"
## [1] "MSE-F: -2.2145"
## [1] "IS R Squared: 0"
## [1] "OOS R Squared: -0.0503"
## [1] "MSE-F: -18.9153"
## [1] "IS R Squared: 5e-04"
## [1] "OOS R Squared: -0.0031"
## [1] "MSE-F: -1.2368"

## [1] "port_S_Q2.csv"
## [1] "S_Q2"
## ##------ Mon Aug  8 10:37:11 2022 ------##
## [1] "OOS R Squared: -0.0385"
## [1] "MSE-F: -14.6964"

## [1] "IS R Squared: 0.0037"
## [1] "OOS R Squared: -0.0238"
## [1] "MSE-F: -9.1721"
## [1] "IS R Squared: 0.0038"
## [1] "OOS R Squared: -0.0022"
## [1] "MSE-F: -0.8541"
## [1] "IS R Squared: 0.0045"
## [1] "OOS R Squared: -0.0022"
## [1] "MSE-F: -0.8793"
## [1] "IS R Squared: 0.0049"
## [1] "OOS R Squared: -0.0301"
## [1] "MSE-F: -11.5508"
## [1] "IS R Squared: 3e-04"
## [1] "OOS R Squared: -0.0114"
## [1] "MSE-F: -4.4504"

## [1] "port_S_Q3.csv"
## [1] "S_Q3"
## ##------ Mon Aug  8 10:37:16 2022 ------##
## [1] "OOS R Squared: -0.0146"
## [1] "MSE-F: -5.6887"

## [1] "IS R Squared: 0.0062"
## [1] "OOS R Squared: -0.022"
## [1] "MSE-F: -8.515"
## [1] "IS R Squared: 0.0119"
## [1] "OOS R Squared: 0.0015"
## [1] "MSE-F: 0.5885"
## [1] "IS R Squared: 0.0134"
## [1] "OOS R Squared: 3e-04"
## [1] "MSE-F: 0.1206"
## [1] "IS R Squared: 0.0077"
## [1] "OOS R Squared: -0.0286"
## [1] "MSE-F: -10.9892"
## [1] "IS R Squared: 0.0012"
## [1] "OOS R Squared: -0.0028"
## [1] "MSE-F: -1.0926"

## [1] "port_S_Q4.csv"
## [1] "S_Q4"
## ##------ Mon Aug  8 10:37:20 2022 ------##
## [1] "OOS R Squared: -0.002"
## [1] "MSE-F: -0.7863"

## [1] "IS R Squared: 0.0027"
## [1] "OOS R Squared: -0.0108"
## [1] "MSE-F: -4.2297"
## [1] "IS R Squared: 0.0042"
## [1] "OOS R Squared: -4e-04"
## [1] "MSE-F: -0.1633"
## [1] "IS R Squared: 0.0058"
## [1] "OOS R Squared: 0"
## [1] "MSE-F: -0.0176"
## [1] "IS R Squared: 0.0045"
## [1] "OOS R Squared: -0.0155"
## [1] "MSE-F: -6.033"
## [1] "IS R Squared: 3e-04"
## [1] "OOS R Squared: -0.0094"
## [1] "MSE-F: -3.6644"

## [1] "port_S_Q5.csv"
## [1] "S_Q5"
## ##------ Mon Aug  8 10:37:24 2022 ------##
## [1] "OOS R Squared: -0.0056"
## [1] "MSE-F: -2.2192"

## [1] "IS R Squared: 0.003"
## [1] "OOS R Squared: -0.0154"
## [1] "MSE-F: -5.9982"
## [1] "IS R Squared: 0.0034"
## [1] "OOS R Squared: -0.0094"
## [1] "MSE-F: -3.6863"
## [1] "IS R Squared: 0.0047"
## [1] "OOS R Squared: -0.0113"
## [1] "MSE-F: -4.4283"
## [1] "IS R Squared: 0.0051"
## [1] "OOS R Squared: -0.0202"
## [1] "MSE-F: -7.8369"
## [1] "IS R Squared: 5e-04"
## [1] "OOS R Squared: -0.0059"
## [1] "MSE-F: -2.3111"

Table 2 - Forecasts of portfolio returns

This table demonstrates the in-sample and out-of-sample R-squares for the market and six size and book-to-market equity ratio sorted portfolios from predictive regressions and the Sum-of-the-Parts method. IS R-squares are estimated using the whole sample period and the OOS R-squares are calculated compare the forecast error of the model against the historical mean model. The full sample period starts from Feb 1966 to December 2019 and the IS period is set to be 20 years with forecsats beginning in Feb 1986. The MSE-F statistics are calculated to test the hypothesis \(H_0: \text{out-of-sample R-squares} = 0\) vs \(H_1: \text{out-of-sample R-squares} \neq 0\).

Predictors here are all in log terms.

gt(table2.df, rowname_col = "rowname", groupname_col = "portname") %>%
  tab_header(title = "Table 2 - Forecasts of portfolio returns",
             subtitle = paste(freq_name(freq = freq), " data starts from ", first(data_decompose$month), " and ends in ", last(data_decompose$month), ".", sep = "")) %>%
  fmt_number(columns = 1:4, decimals = 6, suffixing = TRUE)

Table 2 - Forecasts of portfolio returns
monthly data starts from Jan 1967 and ends in Dec 2019.
	IS_r.squared	OOS_r.squared	MAE_A	MSE_F	McCracken
Market
DP	0.010630	−0.010370	0.032611	−4.054026
PE	0.004546	−0.002340	0.032475	−0.922206
EY	0.005117	−0.001669	0.032494	−0.657989
DY	0.011952	−0.008557	0.032621	−3.351461
Payout	0.000117	−0.007919	0.032081	−3.103538
SOP	NA	0.004656	0.032173	1.852440	**
B_Q1
DP	0.003430	−0.026455	0.033723	−10.180522
PE	0.008801	0.006246	0.033457	2.482527	**
EY	0.009493	0.007637	0.033437	3.039967	**
DY	0.004117	−0.022103	0.033709	−8.542070
Payout	0.002513	−0.004477	0.033401	−1.760410
SOP	NA	0.003427	0.033586	1.361804	*
B_Q2
DP	0.008966	−0.018816	0.029656	−7.294867
PE	0.003235	−0.000583	0.028720	−0.230044
EY	0.002954	−0.000488	0.028713	−0.192759
DY	0.008541	−0.015357	0.029546	−5.974138
Payout	0.001761	−0.014102	0.028886	−5.492749
SOP	NA	0.007234	0.028854	2.885614	**
B_Q3
DP	0.008213	−0.003429	0.029910	−1.349822
PE	0.004521	0.002249	0.029867	0.890414	*
EY	0.004795	0.003114	0.029850	1.233967	*
DY	0.008665	−0.001131	0.029868	−0.446121
Payout	0.002044	−0.004448	0.029712	−1.749093
SOP	NA	0.004202	0.029832	1.671079	**
B_Q4
DP	0.001832	−0.011969	0.032358	−4.671837
PE	0.004903	−0.007702	0.032167	−3.019209
EY	0.005551	−0.006389	0.032147	−2.507720
DY	0.002242	−0.011676	0.032368	−4.558814
Payout	0.000466	−0.004344	0.031977	−1.708440
SOP	NA	−0.002544	0.032221	−1.004716
B_Q5
DP	0.003353	−0.011880	0.040577	−4.637465
PE	0.006982	−0.008524	0.040589	−3.338611
EY	0.007776	−0.007548	0.040615	−2.959140
DY	0.004024	−0.012368	0.040667	−4.825495
Payout	0.000833	−0.004619	0.040022	−1.816165
SOP	NA	0.006754	0.040072	2.692613	**
S_Q1
DP	0.000101	−0.040194	0.048213	−15.263039
PE	0.000248	−0.005411	0.047418	−2.125673
EY	0.000672	−0.005638	0.047358	−2.214482
DY	0.000003	−0.050295	0.048243	−18.915299
Payout	0.000544	−0.003141	0.047536	−1.236841
SOP	NA	−0.247861	0.052651	−78.657072
S_Q2
DP	0.003674	−0.023773	0.038096	−9.172123
PE	0.003770	−0.002167	0.037204	−0.854053
EY	0.004541	−0.002231	0.037161	−0.879346
DY	0.004869	−0.030124	0.038279	−11.550840
Payout	0.000274	−0.011395	0.037658	−4.450400
SOP	NA	−0.038543	0.038076	−14.696445
S_Q3
DP	0.006191	−0.022032	0.036441	−8.514969
PE	0.011867	0.001488	0.035960	0.588508
EY	0.013353	0.000305	0.036007	0.120606
DY	0.007689	−0.028617	0.036741	−10.989161
Payout	0.001228	−0.002774	0.035929	−1.092587
SOP	NA	−0.014575	0.036030	−5.688693
S_Q4
DP	0.002720	−0.010824	0.039663	−4.229732
PE	0.004155	−0.000413	0.039350	−0.163254
EY	0.005813	−0.000045	0.039361	−0.017590
DY	0.004460	−0.015510	0.039772	−6.032989
Payout	0.000294	−0.009364	0.039532	−3.664391
SOP	NA	−0.001989	0.039340	−0.786274
S_Q5
DP	0.002967	−0.015420	0.049478	−5.998235
PE	0.003446	−0.009420	0.049118	−3.686322
EY	0.004731	−0.011338	0.049236	−4.428343
DY	0.005054	−0.020242	0.049866	−7.836933
Payout	0.000494	−0.005885	0.048822	−2.311143
SOP	NA	−0.005636	0.048550	−2.219166

Figure 4 - Monthly return predictions

Here I only present the monthly predictions of the historical mean model, the SOP method and the predictive regressions based on the dividend-price ratio and the earnings-price ratio.

## [1] "market_Allfirms.csv"
## [1] "Market"
## ##------ Mon Aug  8 10:37:33 2022 ------##

## [1] "port_B_Q1.csv"
## [1] "B_Q1"
## ##------ Mon Aug  8 10:37:39 2022 ------##

## [1] "port_B_Q2.csv"
## [1] "B_Q2"
## ##------ Mon Aug  8 10:37:43 2022 ------##

## [1] "port_B_Q3.csv"
## [1] "B_Q3"
## ##------ Mon Aug  8 10:37:47 2022 ------##

## [1] "port_B_Q4.csv"
## [1] "B_Q4"
## ##------ Mon Aug  8 10:37:51 2022 ------##

## [1] "port_B_Q5.csv"
## [1] "B_Q5"
## ##------ Mon Aug  8 10:37:57 2022 ------##

## [1] "port_S_Q1.csv"
## [1] "S_Q1"
## ##------ Mon Aug  8 10:38:01 2022 ------##

## [1] "port_S_Q2.csv"
## [1] "S_Q2"
## ##------ Mon Aug  8 10:38:07 2022 ------##

## [1] "port_S_Q3.csv"
## [1] "S_Q3"
## ##------ Mon Aug  8 10:38:12 2022 ------##

## [1] "port_S_Q4.csv"
## [1] "S_Q4"
## ##------ Mon Aug  8 10:38:15 2022 ------##

## [1] "port_S_Q5.csv"
## [1] "S_Q5"
## ##------ Mon Aug  8 10:38:20 2022 ------##

Figure 5 - Trading Performance (with no trading restrictions)

• Markowitz optimal weight on the risky portfolio using the SOP method and the historical mean model.
• Markowitz optimal weight on the risky portfolio using all models.
• Out-of-sample performance of the trading strategies - the SOP method, the historical mean model and the buy-and-hold strategy - with no trading restrictions.

## Warning in window.default(x, ...): 'start' value not changed

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## Warning in xy.coords(x = matrix(rep.int(tx, k), ncol = k), y = x, log = log, :
## 786 y values <= 0 omitted from logarithmic plot

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## Warning in xy.coords(x = matrix(rep.int(tx, k), ncol = k), y = x, log = log, :
## 387 y values <= 0 omitted from logarithmic plot

## Warning in window.default(x, ...): 'start' value not changed

## Warning in window.default(x, ...): 'start' value not changed

## Warning in xy.coords(x = matrix(rep.int(tx, k), ncol = k), y = x, log = log, :
## 387 y values <= 0 omitted from logarithmic plot

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## Warning in window.default(x, ...): 'start' value not changed

Table 3 - Certaint equivalent gains

Trading Strategies: certaint equivalent gains

This table shows the out-of-sample portfolio choice results at monthly frequencies from predictive regressions and the SOP method. The trading strategy for each portfolio is designed by optimally allocating funds between the risk-free asset and the corresponding risky portfolio. The certainty equivalent return is \(\overline{rp} - \frac{1}{2} \gamma \hat{\sigma}_{rp}^{2}\) with a risk-aversion coefficient \(\gamma = 3\). The annualised certainty equivalent gain (in percentage) is the monthly certainty equivalent gain multiplied by the corresponding frequency (e.g. 12 for monthly data).

dt <- table3.df %>%
  filter(rowname %in% c(ratio_names, "sop_simple")) %>%
  select(CEGs_annualised, rowname, portname)

as.data.frame(matrix(dt$CEGs_annualised, byrow = F, nrow = length(ratio_names) + 1, ncol = length(id.names))) %>%
  `colnames<-`(unique(dt$portname)) %>%
  mutate(Variable = unique(dt$rowname)) %>%
  # round(digits = 4) %>%
  as.tbl() %>%
  select(Variable, unique(dt$portname)) %>%
  gt(rowname_col = "Variable") %>%
  tab_header(title = "Table 3 - Trading Strategies: certainty equivalent gains",
             subtitle = paste(str_to_title(freq_name(freq = freq)), " data starts from ", first(data_decompose$month) + 20, " and ends in ", last(data_decompose$month), ".", sep = "")) %>%
  fmt_percent(columns = 2:(length(id.names)+1), decimals = 2)

Table 3 - Trading Strategies: certainty equivalent gains
Monthly data starts from Jan 1987 and ends in Dec 2019.
	Market	B_Q1	B_Q2	B_Q3	B_Q4	B_Q5	S_Q1	S_Q2	S_Q3	S_Q4	S_Q5
sop_simple	1.10%	5,965.60%	3.56%	1.72%	0.21%	0.73%	−66.08%	−19.16%	2.61%	1.03%	0.42%
DP	−2.85%	−1,462.50%	−2.99%	0.73%	−2.74%	−3.51%	−94.70%	3.51%	5.83%	2.95%	−4.88%
PE	0.84%	6,701.91%	0.39%	2.01%	−1.32%	−2.59%	−7.49%	−8.86%	1.88%	1.64%	−0.95%
EY	0.98%	6,761.86%	0.28%	2.18%	−1.02%	−2.72%	−8.65%	−11.60%	1.20%	1.78%	−2.58%
DY	−2.36%	1,846.85%	−1.74%	1.20%	−2.82%	−3.60%	−134.88%	1.84%	4.58%	1.51%	−10.76%
Payout	−4.13%	−6,716.99%	−20.40%	−1.97%	−0.93%	−0.32%	−0.88%	3.52%	0.02%	−2.93%	−0.97%

Table 4 - Sharpe ratio Gains

Trading Strategies: Sharpe ratio Gains

This table presents the Sharpe ratio of the out-of-sample performance of trading strategies, allocating funds between risk-free and risky assets for each portfolio. The annualised Sharpe ratio is generated by multipling the monthly Sharpe ratio by square root of the corresponding frequency (e.g. \(\sqrt{12}\) for monthly data).

dt <- table4.df %>%
  filter(rowname %in% c(ratio_names, "sop_simple")) %>%
  select(SRG_annualised, rowname, portname)

as.data.frame(matrix(dt$SRG_annualised, byrow = F, nrow = length(ratio_names) + 1, ncol = length(id.names))) %>%
  `colnames<-`(unique(dt$portname)) %>%
  mutate(Variable = unique(dt$rowname)) %>%
  # round(digits = 4) %>%
  as.tbl() %>%
  select(Variable, unique(dt$portname)) %>%
  gt(rowname_col = "Variable") %>%
  tab_header(title = "Table 4 - Trading Strategies: Sharpe ratio gains", 
             subtitle = paste(str_to_title(freq_name(freq = freq)), " data starts from ", first(data_decompose$month) + 20, " and ends in ", last(data_decompose$month), ".", sep = "")) %>%
  fmt_number(columns = 2:(length(id.names)+1), decimals = 4) 

Table 4 - Trading Strategies: Sharpe ratio gains
Monthly data starts from Jan 1987 and ends in Dec 2019.
	Market	B_Q1	B_Q2	B_Q3	B_Q4	B_Q5	S_Q1	S_Q2	S_Q3	S_Q4	S_Q5
sop_simple	0.0524	0.0404	0.1240	0.0513	0.0208	0.0604	−0.1259	−0.0074	0.1385	0.0860	0.1167
DP	−0.1667	0.3357	−0.1378	−0.0265	−0.1652	−0.2850	−0.4922	0.1314	0.1699	0.0866	0.0164
PE	0.0674	0.4973	−0.0245	0.0511	−0.0646	−0.2148	−0.1389	−0.1499	0.0699	0.0394	0.0439
EY	0.0827	0.7366	−0.0268	0.0603	−0.0386	−0.2299	−0.1358	−0.1697	0.0645	0.0532	0.0481
DY	−0.1441	0.3389	−0.1021	0.0013	−0.1710	−0.2927	−0.5107	0.1366	0.1460	0.0571	0.0250
Payout	−0.1547	−0.0071	−0.1644	−0.0573	−0.0433	−0.0355	−0.0189	0.0573	−0.0096	−0.0925	−0.0912

Figure 6 - Sensitivity of Certainty Equivalent Gains relative to Risk-Aversion level

This figure presents the out-of-sample portfolio choice results at monthly frequency from bivariate predictive regressions and the SOP method with different levels of risk-aversion. To show that our previous results hold with respect to investors with different levels of risk aversion, we evaluate the changes in certainty equivalent gains with respect to the changes in the level of risk-aversion. The results of the trading strategy reported here are without trading restrictions (as in Table 5), allocating funds between the risk-free asset and the risky equity portfolio. The portfolio choice results are evaluated in the certainty equivalent return with relative risk-aversion coefficient \(\gamma\), with ${\(0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5\)}$. Risky equity portfolios include the market portfolio and six size and book-to-market equity sorted portfolios, BH, BM, BL, SH, SM and SL. The annualised certainty equivalent gain is the monthly certainty equivalent gain multiplied by twelve. The sample period is from February 1966 to December 2019 and the out-of-sample period starts in March 1986.

## [1] "Market"
## ##------ Mon Aug  8 10:38:35 2022 ------##
## [1] "B_Q1"
## ##------ Mon Aug  8 10:38:35 2022 ------##
## [1] "B_Q2"
## ##------ Mon Aug  8 10:38:36 2022 ------##
## [1] "B_Q3"
## ##------ Mon Aug  8 10:38:36 2022 ------##
## [1] "B_Q4"
## ##------ Mon Aug  8 10:38:36 2022 ------##
## [1] "B_Q5"
## ##------ Mon Aug  8 10:38:36 2022 ------##
## [1] "S_Q1"
## ##------ Mon Aug  8 10:38:36 2022 ------##
## [1] "S_Q2"
## ##------ Mon Aug  8 10:38:36 2022 ------##
## [1] "S_Q3"
## ##------ Mon Aug  8 10:38:36 2022 ------##
## [1] "S_Q4"
## ##------ Mon Aug  8 10:38:37 2022 ------##
## [1] "S_Q5"
## ##------ Mon Aug  8 10:38:37 2022 ------##

## Warning: Removed 13 rows containing missing values (geom_point).

## Warning: Removed 13 row(s) containing missing values (geom_path).

Table 5 - MSPE-adjusted Statistic

MSPE-adjusted Statistic

This table presents the MSEP-adjusted Statistics, evaluating the statistical significance of the out-of-sample R-squared statistics of each model in the corresponding portfolio.

See Rapach et al., (2010) and Clark and West (2007) for the detailed procedure.

table5.df <- data.frame()
for (port in names(TABLE5)) {
  pt <- TABLE5[[port]]
  pt$rowname <- rownames(pt)
  pt$portname <- port
  colnames(pt)[4] <- "star"
  table5.df <- rbind.data.frame(table5.df, pt)
}

table5.output <- gt(table5.df, rowname_col = "rowname", groupname_col = "portname") %>%
  fmt_percent(columns = vars(OOS_r.squared, mspe_pvalue), decimals = 2) %>%
  fmt_number(columns = vars(mspe_t), decimals = 4) %>%
  tab_header(title = "Table 5 - MSPE-adjusted Statistic",
             subtitle = paste(str_to_title(freq_name(freq = freq)), " data starts from ", first(data_decompose$month), " and ends in ", last(data_decompose$month), ".", sep = ""))

## Warning: `columns = vars(...)` has been deprecated in gt 0.3.0:
## * please use `columns = c(...)` instead

## Warning: `columns = vars(...)` has been deprecated in gt 0.3.0:
## * please use `columns = c(...)` instead

## Warning: `columns = vars(...)` has been deprecated in gt 0.3.0:
## * please use `columns = c(...)` instead

## Warning: `columns = vars(...)` has been deprecated in gt 0.3.0:
## * please use `columns = c(...)` instead

table5.output

Table 5 - MSPE-adjusted Statistic
Monthly data starts from Jan 1967 and ends in Dec 2019.
	OOS_r.squared	mspe_t	mspe_pvalue	star
Market
DP	−1.04%	0.7486	22.73%
PE	−0.23%	0.6344	26.31%
EY	−0.17%	0.7180	23.66%
DY	−0.86%	0.9945	16.03%
Payout	−0.79%	−1.4743	92.94%
SOP	0.47%	1.2007	11.53%
B_Q1
DP	−2.65%	−1.2301	89.03%
PE	0.62%	1.5904	5.63%	*
EY	0.76%	1.7027	4.47%	**
DY	−2.21%	−0.8657	80.64%
Payout	−0.45%	−0.2789	60.98%
SOP	0.34%	1.3113	9.53%	*
B_Q2
DP	−1.88%	1.0135	15.57%
PE	−0.06%	0.4811	31.54%
EY	−0.05%	0.4245	33.57%
DY	−1.54%	1.0565	14.57%
Payout	−1.41%	0.1862	42.62%
SOP	0.72%	1.6587	4.90%	**
B_Q3
DP	−0.34%	1.1380	12.79%
PE	0.22%	0.9720	16.58%
EY	0.31%	1.0429	14.88%
DY	−0.11%	1.2630	10.37%
Payout	−0.44%	0.2325	40.81%
SOP	0.42%	1.4487	7.41%	*
B_Q4
DP	−1.20%	−0.6481	74.14%
PE	−0.77%	−0.0888	53.54%
EY	−0.64%	0.0955	46.20%
DY	−1.17%	−0.4845	68.58%
Payout	−0.43%	−0.5897	72.21%
SOP	−0.25%	−0.2050	58.12%
B_Q5
DP	−1.19%	−0.1263	55.02%
PE	−0.85%	0.3902	34.83%
EY	−0.75%	0.5291	29.85%
DY	−1.24%	0.1563	43.79%
Payout	−0.46%	−0.4577	67.63%
SOP	0.68%	1.7225	4.29%	**
S_Q1
DP	−4.02%	−1.4667	92.84%
PE	−0.54%	−1.1431	87.32%
EY	−0.56%	−0.8262	79.54%
DY	−5.03%	−1.3474	91.07%
Payout	−0.31%	−0.6456	74.05%
SOP	−24.79%	−0.9817	83.66%
S_Q2
DP	−2.38%	0.6422	26.06%
PE	−0.22%	0.5428	29.38%
EY	−0.22%	0.7202	23.59%
DY	−3.01%	0.9764	16.47%
Payout	−1.14%	−0.4136	66.03%
SOP	−3.85%	0.7619	22.33%
S_Q3
DP	−2.20%	1.0221	15.37%
PE	0.15%	1.5357	6.27%	*
EY	0.03%	1.6720	4.77%	**
DY	−2.86%	1.2616	10.39%
Payout	−0.28%	−0.1373	55.46%
SOP	−1.46%	0.9986	15.93%
S_Q4
DP	−1.08%	0.2188	41.35%
PE	−0.04%	0.7316	23.24%
EY	−0.00%	1.0382	14.99%
DY	−1.55%	0.5547	28.97%
Payout	−0.94%	−1.7303	95.78%
SOP	−0.20%	0.3719	35.51%
S_Q5
DP	−1.54%	0.7119	23.85%
PE	−0.94%	0.7428	22.90%
EY	−1.13%	1.0219	15.37%
DY	−2.02%	1.0662	14.35%
Payout	−0.59%	−1.1132	86.68%
SOP	−0.56%	0.3033	38.09%